L(s) = 1 | + (0.5 + 0.866i)2-s + (0.340 − 0.196i)3-s + (−0.499 + 0.866i)4-s − 1.77i·5-s + (0.340 + 0.196i)6-s + (2.18 + 1.26i)7-s − 0.999·8-s + (−1.42 + 2.46i)9-s + (1.53 − 0.888i)10-s + (2.98 − 1.72i)11-s + 0.393i·12-s + (2.99 − 2.01i)13-s + 2.52i·14-s + (−0.349 − 0.604i)15-s + (−0.5 − 0.866i)16-s + (4.08 + 0.551i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.196 − 0.113i)3-s + (−0.249 + 0.433i)4-s − 0.794i·5-s + (0.138 + 0.0802i)6-s + (0.824 + 0.476i)7-s − 0.353·8-s + (−0.474 + 0.821i)9-s + (0.486 − 0.280i)10-s + (0.899 − 0.519i)11-s + 0.113i·12-s + (0.830 − 0.557i)13-s + 0.673i·14-s + (−0.0901 − 0.156i)15-s + (−0.125 − 0.216i)16-s + (0.991 + 0.133i)17-s + ⋯ |
Λ(s)=(=(442s/2ΓC(s)L(s)(0.748−0.663i)Λ(2−s)
Λ(s)=(=(442s/2ΓC(s+1/2)L(s)(0.748−0.663i)Λ(1−s)
Degree: |
2 |
Conductor: |
442
= 2⋅13⋅17
|
Sign: |
0.748−0.663i
|
Analytic conductor: |
3.52938 |
Root analytic conductor: |
1.87866 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ442(237,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 442, ( :1/2), 0.748−0.663i)
|
Particular Values
L(1) |
≈ |
1.79197+0.680357i |
L(21) |
≈ |
1.79197+0.680357i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.5−0.866i)T |
| 13 | 1+(−2.99+2.01i)T |
| 17 | 1+(−4.08−0.551i)T |
good | 3 | 1+(−0.340+0.196i)T+(1.5−2.59i)T2 |
| 5 | 1+1.77iT−5T2 |
| 7 | 1+(−2.18−1.26i)T+(3.5+6.06i)T2 |
| 11 | 1+(−2.98+1.72i)T+(5.5−9.52i)T2 |
| 19 | 1+(3.66−6.34i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−0.147+0.0854i)T+(11.5−19.9i)T2 |
| 29 | 1+(−3.57+2.06i)T+(14.5−25.1i)T2 |
| 31 | 1+9.12iT−31T2 |
| 37 | 1+(9.10−5.25i)T+(18.5−32.0i)T2 |
| 41 | 1+(−0.111+0.0644i)T+(20.5−35.5i)T2 |
| 43 | 1+(2.65−4.60i)T+(−21.5−37.2i)T2 |
| 47 | 1+2.05T+47T2 |
| 53 | 1+4.26T+53T2 |
| 59 | 1+(−0.0723+0.125i)T+(−29.5−51.0i)T2 |
| 61 | 1+(13.4+7.76i)T+(30.5+52.8i)T2 |
| 67 | 1+(−4.08−7.07i)T+(−33.5+58.0i)T2 |
| 71 | 1+(4.14+2.39i)T+(35.5+61.4i)T2 |
| 73 | 1+9.40iT−73T2 |
| 79 | 1−2.76iT−79T2 |
| 83 | 1−6.67T+83T2 |
| 89 | 1+(7.64+13.2i)T+(−44.5+77.0i)T2 |
| 97 | 1+(5.50+3.17i)T+(48.5+84.0i)T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.43096191962163195634108880430, −10.34678323896907466250882086620, −9.013410005463232474937120423575, −8.136787239863380331361326403610, −8.056590718324234691075064696791, −6.26980596933178375449487035937, −5.57017860929858916966997861734, −4.61564750928493783209618682902, −3.38230867010125758032690870366, −1.58808483659907408427536002687,
1.41760040079499472131143723841, 3.00444931019149873601740230754, 3.92965288864730175314098993390, 4.99179078441099472500891390661, 6.45607942835437656949293215018, 7.05344880576985201451818115336, 8.600285309802190125740947008326, 9.195042868796925179992049378272, 10.44026725338370034057808732940, 10.98598226643357797152079967958